BENDING STRESSES IN BEAMS

4.1 SIMPLE BENDING OR PURE

BENDING

When some external force acts on a beam, the

shear force and bending moments are set up at

all the sections of the beam

Due to shear force and bending moment, the

beam undergoes deformation. The material of the

beam offers resistance to deformation

Stresses introduced by bending moment are

known as bending stresses

Bending stresses are indirect normal stresses

to pure bending is called

4.1 SIMPLE BENDING OR PURE

BENDING

When a length of a beam is subjected to zero

shear force and constant bending moment, then

that length of beam is subjected to pure bending

or simple pending.

The stress set up in that length of the beam due

simple bending stresses

SIMPLE BENDING OR PURE

BENDING

Consider a simply supported beam with over

hanging portions of equal lengths. Suppose the

beam is subjected to equal loads of intensity W at

either ends of the over hanging portion

In the portion of beam of length l, the beam is

subjected to constant bending moment of

intensity w x a and shear force in this portion is

zero

Hence the portion AB is said to be subjected to

pure bending or simple bending

ASSUMPTIONS FOR THE

THEORY OF PURE BENDING

The material of the beam is isotropic and

homogeneous. Ie of same density and elastic

properties throughout

The beam is initially straight and unstressed and

all the longitudinal filaments bend into a circular

arc with a common centre of curvature

The elastic limit is nowhere exceeded during the

bending

Young's modulus for the material is the same in

tension and compression

ASSUMPTIONS FOR THE

THEORY OF PURE BENDING

The transverse sections which were plane before

bending remain plane after bending also

Radius of curvature is large compared to the

dimensions of the cross section of the beam

There is no resultant force perpendicular to any

cross section

All the layers of the beam are free to elongate

and contract, independently of the layer, above or

below it.

to the axis of the beam and

THEORY OF SIMPLE

BENDING Consider a beam subjected to simple bending.

Consider an infinitesimal element of length dx which is a part of this beam. Consider two transverse sections AB and CD which are normal

parallel to each other.

Due to the bending action the element ABCD is

deformed to A’B’C’D’ (concave curve).

The layers of the beam are not of the same length before bending and after bending .

4.3 THEORY OF SIMPLE

BENDING

The layer AC is shortened to A’C’. Hence it is

subjected to compressive stress

The layer BD is elongated to B’D’. Hence it is

subjected to tensile stresses.

Hence the amount of shortening decrease from

the top layer towards bottom and the amount of

elongation decreases from the bottom layer

towards top

Therefore there is a layer in between which

neither elongates nor shortens. This layer is

called neutral layer .

THEORY OF SIMPLE

BENDING

The filaments/ fibers of the material are

subjected to neither compression nor to tension

The line of intersection of the neutral layer with

transverse section is called neutral axis (N-N).

Hence the theory of pure bending states that the

amount by which a layer in a beam subjected to

pure bending, increases or decreases in length,

depends upon the position of the layer w.r.t

neutral axis N-N.

EXPRESSION FOR BENDING

STRESS

Consider a beam subjected to simple bending.

Consider an infinitesimal element of length dx

which is a part of this beam. Consider two

transverse sections AB and CD which are normal

to the axis of the beam. Due to the bending action

the element ABCD is deformed to A’B’C’D’

(concave curve).

The lines B’A’ and D’C’ when extended meet at

point O (which is the centre of curvature for the

circular arc formed).

Let R be the radius of curvature of the neutral

axis.

E’F’= (R + y) x θ

STRAIN VARIATION ALONG THE

DEPTH OF BEAM

Consider a layer EF at a distance y from the

neutral axis. After bending this layer will be

deformed to E’F’.

Strain developed= (E’F’-EF)/EF

EF=NN=dx=R x θ

4.4.1 STRAIN VARIATION ALONG

THE DEPTH OF BEAM

Strain developed eb= { (R + y) x θ - R x θ)}/ R x

θ=y/R

STRESS VARIATION WITH DEPTH OF BEAM

σ/E= y/R or σ= Ey/R or σ/y = E/R

Hence σ varies linearly with y (distance from

neutral axis)

Therefore stress in any layer is directly

proportional to the distance of the layer from the

neutral layer

the neutral axis is given by σ/y

NEUTRAL AXIS

For a beam subjected to a pure bending moment,

the stresses generated on the neutral layer is

zero.

Neutral axis is the line of intersection of neutral

layer with the transverse section

Consider the cross section of a beam subjected

to pure bending. The stress at a distance y from

=E/R

4.5 NEUTRAL AXIS

σ= E x y/R;

The force acting perpendicular to this section,

dF= E x y/R x dA, where dA is the cross sectional

area of the strip/layer considered.

Pure bending theory is based on an assumption

that “There is no resultant force perpendicular to

any cross section”. Hence F=0;

Hence, E/R x ∫ydA=0

=> ∫ydA= Moment of area of the entire cross

section w.r.t the neutral axis=0

ntroidal axis gives the p

NEUTRAL AXIS

Moment of area of any surface w.r.t the centroidal

axis is zero. Hence neutral axis and centroidal

axis for a beam subjected to simple bending are

the same.

Neutral axis coincides with centrodial axis or the

ce osition of neutral axis

MOMENT OF RESISTANCE

Due to the tensile and compressive stresses,

forces are exerted on the layers of a beam

subjected to simple bending

These forces will have moment about the neutral

axis. The total moment of these forces about the

neutral axis is known as moment of resistance of

that section

We have seen that force on a layer of cross

sectional area dA at a distance y from the neutral

axis,

dF= (E x y x dA)/R

Moment of force dF about the neutral axis= dF x

y= (E x y x dA)/R x y= E/R x (y²dA)

MOMENT OF RESISTANCE Hence the total moment of force about the neutral

axis= Bending moment applied= ∫ E/R x (y²dA)= E/R x Ixx; Ixx is the moment of area about the neutral axis/centroidal axis.

Hence M=E/R x Ixx

Or M/Ixx=E/R

Hence M/Ixx=E/R = σb/y;σb is also known as flexural stress (Fb)

Hence M/Ixx=E/R=Fb/y

The above equation is known as bending equation

This can be remembered using the sentence “Elizabeth Rani May I Follow You”

CONDITION OF SIMPLE

BENDING & FLEXURAL RIGIDITY

Bending equation is applicable to a beam

subjected to pure/simple bending. Ie the bending

moment acting on the beam is constant and the

shear stress is zero

However in practical applications, the bending

moment varies from section to section and the

shear force is not zero

But in the section where bending moment is

maximum, shear force (derivative of bending

moment) is zero

Hence the bending equation is valid for the

section where bending moment is maximum

4.7 CONDITION OF SIMPLE

BENDING & FLEXURAL RIGIDITY

Or in other words, the condition of simple

bending may be satisfied at a section where

bending moment is maximum.

Therefore beams and structures are designed

using bending equation considering the section of

maximum bending moment

Flexural rigidity/Flexural resistance of a beam:-

For pure bending of uniform sections, beam will

deflect into circular arcs and for this reason the

term circular bending is often used.

CONDITION OF SIMPLE

BENDING & FLEXURAL RIGIDITY

The radius of curvature to which any beam is

bent by an applied moment M is given by R=EI/M

Hence for a given bending moment, the radius of

curvature is directly related to “EI”

Since radius of curvature is a direct indication of

the degree of flexibility of the beam (larger the

value of R, less flexible the beam is, more rigid

the beam is), EI is known as flexural rigidity of

flexural stiffness of the beam.

The relative stiffnesses of beam sections can

then easily be compared by their EI value

SECTIONAL MODULUS (Z)

Section modulus is defined as the ratio of

moment of area about the centroidal axis/neutral

axis of a beam subjected to bending to the

distance of outermost layer/fibre/filament from the

centroidal axis

Z= Ixx/ymax

From the bending equation, M/Ixx= σbmax/ymax

Hence Ixx/ymax=M/ σbmax

M= σbmax X Z

Higher the Z value for a section, the higher the

BM which it can withstand for a given maximum

stress

ion

VARIOUS SHAPES OR BEAM

SECTIONS

1) For a Rectangular section

Z=Ixx/ymax

Ixx=INA= bd³/12

ymax= d/2

Z= bd²/6

2) For a Rectangular hollow sect

Ixx= 1/12 x (BD³/12 - bd³/12)

Ymax = D/2

Z= (BD³ - bd³)/6D

Z= πD³/32

VARIOUS SHAPES OR BEAM

SECTIONS

3) For a circular section of diameter D,

Ixx= πD^4/64

ymax = D/2

4) For a hollow circular section of outer diameter

D and inner diameter d,

Ina= (πD^4 - d^4)/64

ymax=D/2

Z= (πD^4 - d^4)/32D

BENDING OF FLITCHED

BEAMS

A beam made up of two or more different

materials assumed to be rigidly connected

together and behaving like a single piece is called

a flitched beam or a composite beam.

Consider a wooden beam re-inforced by steel

plates. Let

E1= Modulus of elasticity of steel plate

E2= Modulus of elasticity of wooden beam

M1= Moment of resistance of steel plate

M2= Moment of resistance of wooden beam

4.9 BENDING OF FLITCHED

BEAMS

I1= Moment of inertia of steel plate about neutral

axis

I2= Moment of inertia of wooden beam about

neutral axis.

The bending stresses can be calculated using two

conditions.

Strain developed on a layer at a particular

distance from the neutral axis is the same for

both the materials

Moment of resistance of composite beam is equal

to the sum of individual moment of resistance of

the members.

THANK YOU

4.1 SIMPLE BENDING OR PURE BENDING4.1 SIMPLE BENDING OR PURE BENDING (1)SIMPLE BENDING OR PURE BENDINGASSUMPTIONS FOR THE THEORY OF PURE BENDINGASSUMPTIONS FOR THE THEORY OF PURE BENDING (1)THEORY OF SIMPLE BENDING4.3 THEORY OF SIMPLE BENDINGTHEORY OF SIMPLE BENDING (1)EXPRESSION FOR BENDING STRESS4.4.1 STRAIN VARIATION ALONG THE DEPTH OF BEAM

NEUTRAL AXIS4.5 NEUTRAL AXISNEUTRAL AXIS (1)MOMENT OF RESISTANCEMOMENT OF RESISTANCE (1)CONDITION OF SIMPLE BENDING & FLEXURAL RIGIDITY4.7 CONDITION OF SIMPLE BENDING & FLEXURAL RIGIDITYCONDITION OF SIMPLE BENDING & FLEXURAL RIGIDITY (1)

SECTIONAL MODULUS (Z)VARIOUS SHAPES OR BEAM SECTIONSVARIOUS SHAPES OR BEAM SECTIONS (1)

BENDING OF FLITCHED BEAMS4.9 BENDING OF FLITCHED BEAMSTHANK YOU